Some further remarks on the Ostrowski generalization of Chebyshev's inequality (Q1089119)

The following is a typical result. Let \(p: [a,b]^ 2\to {\mathbb{R}}\) be an integrable function such that \[ \int^{b}_{x}\int^{x}_{a}p(s,t)dt ds=\int^{x}_{a}\int^{b}_{x}p(s,t)dt ds, \] \[ \int^{x}_{a}\int^{b}_{y}p(s,t)dt ds\geq 0\quad (a\leq x\leq y\leq b)\quad and\quad \int^{b}_{x}\int^{y}_{a}p(s,t)dt ds\geq 0\quad (a\leq y\leq x\leq b) \] (or both integrals are nonpositive). If \(F: [a,b]^ 2\to {\mathbb{R}}\) has continuous first partial derivatives and a continuous mixed second partial derivative, then there exist \(\xi\),\(\eta\in [a,b]\) such that \[ \int^{b}_{a}\int^{b}_{a}p(x,y)[F(y,y)-F(x,y)]dx dy=\frac{\partial^ 2F(\xi,\eta)}{\partial x\partial y}\int^{b}_{a}\int^{b}_{a}p(x,y)(y-a)(y-x)dx dy. \] Sum analogues of this and similar results are also presented.